Reduced-Order Modelling of Defect Transport using Surrogate Kinetics: Application to U$_x$Pu$_ {1-x}$N

Technical Summary: Reduced-Order Modelling of Defect Transport using Surrogate Kinetics in UxPu1−xN

Problem Statement
Modeling defect transport in chemically disordered materials, such as mixed actinide nitrides ($U_xPu_{1-x}N$), presents a significant challenge. In chemically uniform lattices, defect migration pathways can be enumerated as a finite set of transition rates. However, in disordered systems, the local chemical environment varies continuously, creating a distribution of migration barriers rather than a closed set. This chemical complexity renders full enumeration of environments via Density Functional Theory (DFT) or Molecular Dynamics (MD) computationally prohibitive. Furthermore, standard continuous-time Markov chain (CTMC) models cannot accommodate the effectively infinite variety of transition rates generated by local disorder. This issue is critical for advanced nuclear fuels like uranium mononitride (UN), which may evolve into $U_xPu_{1-x}N$ solid solutions due to fission or be manufactured as mixed fuels. Predicting fuel performance (swelling, fission gas release, thermal conductivity degradation) requires understanding how compositional variation alters defect energetics and transport, yet direct atomistic simulation across the vast configurational space is infeasible.

Methodology
The authors propose a reduced-order workflow to embed atomic-scale mechanisms into computationally efficient functional forms. The methodology consists of four main stages:

1. Data Generation (Hop-Decorate): Using the Hop-Decorate (HopDec) workflow built on LAMMPS and ASE, the authors generated a database of approximately 3,000 migration barriers ($E_b$) and energy differences ($\Delta E$). This was achieved by identifying prototypical nearest-neighbor (1NN) migration pathways in reference UN structures and "re-decorating" them 1,000 times at global compositions of $U_{0.5}Pu_{0.5}N$, $U_{0.1}Pu_{0.9}N$, and $U_{0.9}Pu_{0.1}N$. Interatomic energies were calculated using a Finnis–Sinclair EAM potential combined with a Buckingham potential, validated against DFT data. The Climbing-Image Nudged Elastic Band (CI-NEB) method was used to evaluate energetics for each configuration.
2. Surrogate Model Construction: The local chemical environment was defined by the counts of Uranium atoms within specific nearest-neighbor (NN) shells around the defect in its initial, final, and saddle-point configurations. Linear surrogate functions were fitted to predict $E_b$ and $\Delta E$ based on these simple descriptors:
   • $\Delta E$ was modeled as a linear function of the change in U counts between minima shells.
   • $E_b$ was modeled as a linear function of U counts in minima and saddle-point shells, plus a species-specific baseline correction ($\beta$).
   • Separate fits were required for U-mediated and Pu-mediated hops on the actinide sublattice due to strong species dependence.
3. Kinetic Monte Carlo (KMC) Simulation: The surrogate models were integrated into a custom lattice-based KMC (laKMC) simulator. Simulations were run on $10 \times 10 \times 10$ nm$^3$ supercells (64,000 atoms) for 100,000 steps. Mean-squared displacements (MSDs) were extracted to calculate long-time diffusivities ($D$) across various temperatures and compositions, with statistical averaging over 50 trajectories per condition.
4. Trapping Analysis: To elucidate the atomic-scale origins of macroscopic transport, the authors analyzed the kinetic-thermodynamic space by calculating disorder-averaged mean residence times ($\langle \tau \rangle_c$) and rate-weighted mean energy changes ($\langle \Delta E \rangle_c$) for each composition.

Key Results
The study analyzed three defect species: actinide vacancies ($V_A$), nitride vacancies ($V_N$), and actinide-nitride divacancies ($V_{A-N}$).

• Migration Energetics: The surrogate models achieved low root-mean-squared errors (RMSE), indicating that migration energetics in $U_xPu_{1-x}N$ are governed primarily by short-range chemical identity rather than complex long-range collective effects.

  • $V_A$ (Actinide Vacancy): Migration is strongly species-controlled. The barrier distribution bifurcates based on whether the migrating atom is U or Pu. The local environment causes only slight broadening; the primary compositional effect is the statistical weighting of U vs. Pu hops.
  • $V_N$ (Nitride Vacancy): Migration exhibits genuine environment-dependent modulation. The barrier distribution evolves continuously with composition, showing multi-lobed features at intermediate compositions corresponding to distinct local actinide configurations. Notably, dilute U in a Pu-rich matrix can create locally favorable, low-barrier pathways.
  • $V_{A-N}$ (Divacancy): The coupled defect shows behavior resembling the sum of isolated vacancies, with migration proceeding via either sublattice. The presence of the pair induces a systematic shift in barrier height but does not fundamentally alter the sensitivity to local coordination.

• Diffusivity Trends:

  • Non-Linearity: Diffusivity does not follow a simple linear interpolation (Vegard's law) between UN and PuN end members.
  • $V_A$ Behavior: Diffusivity shows a sharp, non-linear increase with small additions of Pu. This is driven by the percolation of fast Pu-mediated hops. As Pu concentration increases, the system transitions from being limited by slow U-hops to a network of fast Pu-hops.
  • $V_N$ Behavior: Diffusivity exhibits "sluggish diffusion" at low Pu concentrations, where Pu acts as a kinetic trap, reducing net transport despite the existence of some low-barrier events. At higher Pu concentrations, diffusivity increases as low-barrier pathways become more prevalent and connected.
  • $V_{A-N}$ Behavior: The divacancy diffusivity reflects a competition between the chemical sensitivities of both sublattices, resulting in complex, non-linear composition dependence distinct from isolated vacancies.

• Trapping Analysis: The kinetic-thermodynamic maps ($\langle \tau \rangle_c$ vs. $\langle \Delta E \rangle_c$) revealed that:

  • For $V_A$, increasing Pu content reduces residence time without significantly altering energetic bias, confirming the percolation of fast pathways.
  • For $V_N$, small Pu concentrations increase energetic bias ($\langle \Delta E \rangle_c$) while residence time remains stable, indicating the formation of chemical traps that modulate relative energies rather than just increasing barriers.

Significance and Claims
The paper claims to provide a general, computationally efficient framework for modeling defect transport in chemically disordered materials without resorting to "black-box" machine learning models that lack physical interpretability.

• Reduced-Order Representation: The authors demonstrate that the complex migration energetics of disordered actinide nitrides admit an unexpectedly simple reduced-order representation using linear functions of local coordination counts.
• Mechanistic Insight: The work identifies two dominant mechanisms governing transport: species-controlled kinetics (dominant for actinide vacancies) and environment-dependent trapping (dominant for nitride vacancies). It highlights that macroscopic diffusivity emerges from the competition between barrier distributions, pathway connectivity, and residence times, rather than simply from mean barrier values.
• Scalability: By distilling atomistic data into compact surrogate functions, the approach enables the integration of atomic-scale mechanisms into higher-scale simulations (e.g., cluster dynamics or continuum fuel performance codes). This allows for the efficient modeling of evolving compositions during fuel burnup or transmutation.
• Applicability: While demonstrated on $U_xPu_{1-x}N$, the framework is presented as applicable to other compositionally disordered materials, such as high-entropy alloys and doped semiconductors, where brute-force enumeration of local environments is intractable.

The authors note limitations, including the constraint of divacancies to nearest-neighbor pairs (ignoring potential dissociation/re-association) and the use of empirical potentials that do not explicitly treat electronic effects (e.g., 5f orbital localization in Pu). However, they assert that the current approach successfully bridges the gap between atomistic mechanisms and tractable transport equations for predictive fuel performance modeling.